Question 169960
First we need to find the domain and range of {{{f(x)=6-x^2}}}


Domain: {{{x>=0}}} Given



Range: {{{y<=6}}} (use a graph)



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Finding the Inverse:



{{{f(x)=6-x^2}}} Start with the given function



{{{y=6-x^2}}} Replace f(x) with y



{{{x=6-y^2}}} Switch x and y. The goal is to solve for y



{{{x-6=-y^2}}} Subtract 6 from both sides



{{{-x+6=y^2}}} Multiply both sides by -1



{{{y^2=6-x}}} Rearrange the terms 



{{{y=sqrt(6-x)}}} Take the square root of both sides. Note: since the domain of f(x) is {{{x>=0}}}, this means that we're only considering the positive square root.



So the answer is {{{y=sqrt(6-x)}}} which means that the inverse function is *[Tex \LARGE f^{-1}(x)=\sqrt{6-x}]



Now the domain and range of the inverse is simply the reverse of the domain and range of the original function



So the domain and range of *[Tex \LARGE f^{-1}(x)=\sqrt{6-x}] is:



Domain: {{{x<=6}}}




Range: {{{y>=0}}}